A user's guide to the local arithmetic of hyperelliptic curves

Abstract

A new approach has been recently developed to study the arithmetic of hyperelliptic curves y 2 = f ( x ) $y^2=f(x)$ over local fields of odd residue characteristic via combinatorial data associated to the roots of f $f$ . Since its introduction, numerous papers have used this machinery of ‘cluster pictures’ to compute a plethora of arithmetic invariants associated to these curves. The purpose of this user's guide is to summarise and centralise all of these results in a self-contained fashion, complemented by an abundance of examples.